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Nonsmooth Mathematical Programs with Vanishing Constraints in Banach Spaces

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High-Dimensional Optimization and Probability

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 191))

Abstract

In this chapter, we study the optimization problems with equality, inequality, and vanishing constraints in a Banach space where the objective function and the binding constraints are either differentiable at the optimal solution or Lipschitz near the optimal solution. We derive nonsmooth Karush–Kuhn–Tucker (KKT) type necessary optimality conditions for the above problem where Fréchet (or Gâteaux or Hadamard) derivatives are used for the differentiable functions and the Michel-Penot (M-P) subdifferentials are used for the Lipschitz continuous functions. We also introduce several modifications of some known constraint qualifications like Abadie constraint qualification, Cottle constraint qualification, Slater constraint qualification, Mangasarian–Fromovitz constraint qualification, and linear independence constraint qualification for the above mentioned problem which is called as the nonsmooth mathematical programs with vanishing constraints (NMPVC) in terms of the M-P subdifferentials and establish relationships among them.

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Acknowledgements

The research of Dr. Vivek Laha is supported by UGC-BSR start up grant by University Grant Commission, New Delhi, India (Letter No. F.30-370/2017(BSR)) (Project No. M-14-40). The research of Dr. Vinay Singh is supported by the Science and Engineering Research Board, a statutory body of the Department of Science and Technology (DST), Government of India, through project reference no. EMR/2016/002756. The research of Prof. S.K. Mishra is financially supported by Department of Science and Technology, SERB, New Delhi, India through grant no.: MTR/2018/000121.

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Authors and Affiliations

  1. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India

    Vivek Laha & S. K. Mishra

  2. Department of Mathematics, National Institute of Technology Mizoram, Aizawl, Mizoram, India

    Vinay Singh

  3. Department of Mathematics, Satish Chandra College, Ballia, India

    Yogendra Pandey

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  1. Vivek Laha
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  2. Vinay Singh
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  3. Yogendra Pandey
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  4. S. K. Mishra
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Editors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Ashkan Nikeghbali

  2. Dept. of Industrial & Systems Eng., University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

  3. Moscow Institute of Physics & Technology, Dolgoprudny, Russia

    Andrei M. Raigorodskii

  4. Department of Mathematics and Engineering Sciences, Hellenic Military Academy, Athens, Greece

    Michael Th. Rassias

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Laha, V., Singh, V., Pandey, Y., Mishra, S.K. (2022). Nonsmooth Mathematical Programs with Vanishing Constraints in Banach Spaces. In: Nikeghbali, A., Pardalos, P.M., Raigorodskii, A.M., Rassias, M.T. (eds) High-Dimensional Optimization and Probability. Springer Optimization and Its Applications, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-031-00832-0_13

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